# Inferior Products and Profitable Deception

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Inferior Products and Profitable Deception∗ Paul Heidhues Botond Kőszegi Takeshi Murooka ESMT Central European University University of Munich December 1, 2014 Abstract We analyze conditions facilitating profitable deception in a simple model of a competitive retail market. Firms selling homogenous products set up-front prices that consumers understand and additional prices that naive consumers ignore unless revealed to them by a firm, where we assume that there is a binding floor on the up-front prices. We show that our reduced-form framework captures more elaborate models of the credit-card and mutual-fund markets, and argue that it may apply to bank accounts and mortgages as well. Our main results establish that “bad” products (those with lower social surplus than an alternative) tend to be more reliably profitable than “good” products. Specifically, (1) in a market with a single socially valuable product and sufficiently many firms, a deceptive equilibrium—in which firms hide additional prices—does not exist and firms make zero profits. But perversely, (2) if the product is socially wasteful, then a profitable deceptive equilibrium always exists. Furthermore, (3) in a market with multiple products, since a superior product both diverts sophisticated consumers and renders an inferior product socially wasteful in comparison, it guarantees that firms can profitably sell the inferior product by deceiving consumers. JEL Codes: D14, D18, D21 ∗ The setup of this paper was developed in our earlier working paper “The Market for Deceptive Products,” which we split into two papers. The companion paper is titled “Exploitative Innovation.” Kőszegi thanks the Berkeley Center for Equitable Growth, and Heidhues and Kőszegi thank the European Research Council (Starting Grant #313341) for financial support. We are grateful to Özlem Bedre-Defolie, Stefano DellaVigna, Drew Fudenberg, Michael Grajek, Michael Grubb, Péter Kondor, Nicolas Melissas, Antoinette Schoar, and seminar and conference audiences for useful comments and feedback. 1

1 Introduction In this paper, we investigate circumstances under which firms sell products by deceiving some con- sumers about the products’ full price, focusing (in contrast to much of the literature) on deception that leads to positive equilibrium profits in seemingly competitive industries.1 We identify a novel, perverse aspect of profitable deception: products that generate lower social surplus than the best alternative facilitate deception precisely because they would not survive in the market if consumers understood their full price, and therefore firms often make profits on exactly such bad products but not on good products. We argue that—in seeking these profits—firms disproportionately push bad products and might enter markets for bad products in potentially massive numbers, creating further economic inefficiencies. We develop our insights in the reduced-form model of Section 2.1, which builds on the seminal theory of Gabaix and Laibson (2006). Firms are engaged in simultaneous-move price competition to sell a single homogenous product. Each firm charges a transparent up-front price as well as an additional price, and unless at least one firm (costlessly) unshrouds the additional prices, naive consumers ignore these prices when making purchase decisions. To capture the notion that for some products firms cannot return all profits from later charges by lowering initial charges, we deviate from most existing work and posit that there is a floor on the up-front price.2 We argue that a profitable deceptive equilibrium—wherein all firms shroud additional prices—is the most plausible equilibrium whenever it exists: it is then the unique equilibrium in the variant of our model in which unshrouding carries a cost (no matter how small), and all firms prefer it over an 1 Hidden fees have often enabled firms to reap substantial profits despite seemingly considerable competition, at least at the price-competition stage when entry and marketing costs have been paid and customer bases have been identified and reached. Investigating trade and portfolio data from a large German bank, for example, Hackethal, Inderst and Meyer (2010) document that “bank revenues from security transactions amount to e2,560 per customer per year” (2.4 percent of mean portfolio value), a figure likely well above the marginal cost of serving a customer. Similarly, based on a number of measures, including the 20-percent average premium in interbank purchases of outstanding credit-card balances, Ausubel (1991) argues that credit-card companies make large profits. Ellison and Ellison (2009) describe a variety of obfuscation strategies online computer-parts retailers use, and document that such strategies can generate surprisingly large profits given the near homogeneity of products. These observations, however, do not mean that the net economic surplus taking all operating costs into account are large or even positive in these markets: for example, fixed entry costs can dissipate any profits from the later stage of serving consumers. 2 Armstrong and Vickers (2012), Ko (2012), and Grubb (forthcoming) also analyze models with variants of our price-floor assumption. 1

unshrouded-prices and therefore zero-profit equilibrium. Hence, in our discussion we presume that a deceptive equilibrium is played whenever it exists. In Section 2.2, we establish that our reduced-form framework captures more elaborate models of two important markets that have been invoked as conducive to hidden charges: credit cards and mutual funds. In our model of credit cards, issuers make offers consisting of an up-front price and an interest rate to naive time-inconsistent borrowers, who derive a convenience benefit from using a credit card and underestimate how much interest they will pay. In the case of indifference, each consumer chooses whether to get or charge a card by lexicographically applying an exogenously given preference over cards. Then, any consumer who prefers a firm’s card will get the card if the firm charges an up-front price of zero. Any additional consumers a negative up-front price attracts, therefore, will not use the firm’s card, and hence these consumers are unprofitable. As a result, firms act as if they were facing a price floor of zero. In our model of mutual funds, firms choose front loads investors understand and management fees investors ignore unless explained to them by a firm. We argue that a key section of the Investment Company Act of 1940, which prohibits the dilution of existing investors’ shares in favor of new investors, prevents funds from offering discounts to new consumers, so that the front load must be non-negative. Finally, we briefly and informally lay out several other applications, including mortgages with changing terms, bank accounts, printers, and hotels, emphasizing that in some of these markets there may not be a binding floor on the up-front price. Section 3 presents our basic results. As two benchmark cases, we show that if the price floor is not binding, only a zero-profit deceptive equilibrium exists, and note that if consumers are sophisticated in that they observe and take into account additional prices, firms again earn zero profits. If the price floor is binding and consumers are naive, however, profitable deception may occur. If other firms shroud and the up-front price is at the floor, a firm cannot compete on the up-front price and can compete on the total price only if it unshrouds—but because consumers who learn of the additional prices may not buy the product, the firm may find the latter form of competition unattractive. If this is the case for all firms, a profitable deceptive equilibrium exists. Otherwise, additional prices are unshrouded with probability one, and firms earn zero profits. 2

The above condition for a firm to find unshrouding unattractive has some potentially important implications for when profitable deception occurs. First, if the product is socially wasteful (its value relative to consumers’ outside option is lower than its production cost), a firm that unshrouds cannot go on to profitably sell its product, so no firm ever wants to unshroud. Perversely, therefore, in a socially wasteful industry a profitable deceptive equilibrium always exists. But if the product is socially valuable, a firm that would make sufficiently low profits from deception can earn higher profits from unshrouding and capturing the entire market, so if there is such a firm only a non- deceptive, zero-profit equilibrium exists. Hence, because in an industry with many firms some firm earns low profits, entry into socially valuable industries makes these industries more transparent; and whenever deceptive practices survive in an industry with many firms, our model says that the industry is socially wasteful. We also show that if consumers initially underappreciate the additional price because they are unaware of a valuable add-on they want to spend on, only an unshrouded-prices equilibrium exists. From the perspective of our model, therefore, the observation that unshrouding has not taken place in the credit-card market implies that excessive credit-card borrowing is not only unanticipated, but unwanted by consumers. In Section 4, we extend our model by assuming that there are both sophisticated and naive consumers in the market. Consistent with the predictions of Gabaix and Laibson (2006) and Arm- strong and Vickers (2012) that sophisticated consumers facilitate transparency and efficiency, we find that if the product is socially valuable and there are sufficiently many sophisticated consumers, only a non-deceptive equilibrium exists. In contrast to received wisdom, however, we also show that in a multi-product market with a superior and an inferior product, often sophisticated and naive consumers self-separate into buying the former and the latter product, respectively, and sophis- ticated consumers exert no pressure to unshroud the inferior product’s additional prices. Worse, because the superior product renders the inferior product socially wasteful in relative terms, it guarantees that profitable deception in the market for the inferior product can be maintained. This observation has a striking implication: all it takes for profitable deception to occur in a competitive industry is the existence of an inferior product with a shroudable price component and a binding floor on the up-front price, and firms’ profits derive entirely from selling this inferior product. 3

In Section 5, we turn to extensions and modifications of our framework. We show that if a firm has market power in the superior-product market, it may have an incentive to unshroud to attract naive consumers to itself, especially if—similarly to for instance Vanguard in the mutual- fund market—it sells mostly the superior product. But if the firm’s market power is limited and unshrouding is costly, the extent to which the firm educates consumers is also limited. In Section 6, we discuss some policy implications of and evidence for our model. Since deception is likely to be more common and have more adverse welfare consequences when the price floor is binding, policymakers should concentrate their regulatory efforts on industries with a binding price floor—of which supranormal profits is a telltale sign. We also note that a policymaker can improve outcomes for consumers if she can lower additional prices. And although the overarching prediction of our paper that inferior products facilitate profitable deception and hence should go together with it is difficult to test directly, some auxiliary predictions of our model are consistent with existing evidence. In particular, our framework predicts that firms disproportionately push—for instance through commission-driven intermediaries or persuasive advertising—inferior products and may enter the market for inferior products in mass numbers, and that the regulation of hidden fees may not result in increases in other fees, and hence benefits consumers. In Section 7, we discuss the behavioral-economics and classical literatures most closely related to our paper. While a growing theoretical literature investigates how firms exploit naive consumers by charging hidden or unexpected fees, our paper goes beyond all of the existing work in making the central prediction that socially wasteful and inferior products tend to be sold more profitably than better products, and identifying a number of additional implications of this insight. We conclude in Section 8 by pointing out important further questions raised by our model. 2 Basic Model and Applications 2.1 Setup N ≥ 2 firms compete for naive consumers who value each firm’s product at v > 0 and are looking to buy at most one item. Consumers have an outside option with utility u. Firms play a simultaneous- 4

move game in which they set up-front prices fn and additional prices an ∈ [0, a], and decide whether to costlessly unshroud the additional prices. A consumer who buys a product must pay both prices associated with that product—she cannot avoid the additional price. If all firms shroud, consumers make purchase decisions as if the total price of product n was fn . If at least one firm unshrouds, all firms’ additional prices become known to all consumers, and consumers make purchase decisions based on the true total prices fn + an .3 If consumers weakly prefer buying and are indifferent between a number of firms, each of these firms gets a positive market share, with this share being sn ∈ (0, 1) if consumers are indifferent between all firms. Although we focus most of our analysis and discussion on price misperceptions, we also allow for consumers to misperceive the product’s value, as when investors overestimate the gross return of a managed mutual fund. Paralleling our setup for additional prices, we assume that if all firms shroud, consumers perceive the value of the product to be ṽ, and if at least one firm unshrouds, consumers correctly understand the value to be v. Firm n’s cost of providing the product is cn > 0. We let cmin = minn {cn }, and—to ensure that our industry is competitive in the corresponding classical Bertrand model—assume that cn = cmin for at least two firms n. In addition, we assume that max{v, ṽ + a} > cn for all n; a firm with max{v, ṽ + a} < cn cannot profitably sell its product, so we think of it as not participating in the market. And to avoid uninteresting qualifications, we suppose that v − u 6= cmin . Deviating from much of the literature, we impose a floor on the up-front price: fn ≥ f . We assume that f ≤ min{ṽ, v} − u, which means that whether or not unshrouding occurs, consumers are willing to buy the product if they perceive the total price to be f . We also assume that f ≤ cmin , so that firms are not prevented from setting a zero-profit total price. Our main interest is in studying the Nash-equilibrium outcomes of the above game played between firms. While we fully characterize equilibrium outcomes in our main propositions, in discussing our results we focus on conditions for and properties of deceptive equilibria—equilibria 3 Our assumption on unshrouding—that a single firm can educate all consumers about all additional prices at no cost—is in the context of most real settings unrealistically extreme, especially if (as in our credit-card application) incurring the additional prices depends partly on the consumer’s own behavior. This extreme case is theoretically useful for demonstrating that education often does not happen even if it is very easy. We show in our working paper (Heidhues, Kőszegi and Murooka 2012b) that if unshrouding is somewhat costly or reaches only a fraction of consumers, a deceptive equilibrium is more likely to occur, but our qualitative results do not change. 5

in which all firms shroud additional prices with probability one. Because no firm has an incentive to shroud if at least one firm unshrouds, there is always an unshrouded-prices equilibrium. When a deceptive equilibrium exists, however, it is more plausible than the unshrouded-prices equilibrium for a number of reasons. Most importantly, we show in Appendix A that whenever a deceptive equilibrium exists in our model with no unshrouding cost, it is the unique equilibrium in the variant of our model in which unshrouding carries a positive cost, no matter how small the cost is. In addition, a positive-profit deceptive equilibrium is preferred by all firms to an unshrouded-prices equilibrium. Finally, for the lowest-priced firms, the strategy they play in an unshrouded-prices equilibrium is weakly dominated by the strategy they play in a positive-profit deceptive equilibrium. To simplify our statements, we define the Bertrand outcome as the outcome of a Bertrand price- competition game with rational consumers and transparent prices: consumers buy if and only if v − u > cmin and pay a total price equal to cmin whenever they buy, and firms earn zero profits. 2.2 Applications 2.2.1 Credit Cards Our first economic application is credit cards. We build on a variant of Heidhues and Kőszegi’s (2010) model of a credit market with naive time-inconsistent borrowers.4 Firms and consumers interact over four periods, t = 0, 1, 2, 3. In period t = 0, 1, 2, a consumer aims to maximize total utility ut /β + 3τ =t+1 uτ , where uτ is instantaneous utility in period τ . The short-term discount P factor β is distributed in the population according to the cumulative distribution function Q on [0, 1], where a ≡ maxβ∈(0,1] Q(β)(1/β − 1) exists.5 We posit that the consumer’s period-0 total utility is relevant for welfare evaluations. 4 A substantial body of research in behavioral economics documents that individuals often have a taste for immediate gratification (Laibson, Repetto and Tobacman 2007, Augenblick, Niederle and Sprenger 2013, for instance), that such taste is associated with credit-card borrowing (Meier and Sprenger 2010), and that individuals may be naive about their taste (Laibson et al. 2007, Skiba and Tobacman 2008). And consistent with the specific hypothesis that consumers underestimate costly borrowing on their credit cards, Ausubel (1991) finds that consumers are less responsive to the post-introductory interest rate in credit-card solicitations than to the teaser rate, even though the former is more important in determining the amount of interest they will pay. 5 For presentational simplicity, we use a specification in which β divides current utility instead of the (behav- iorally equivalent) more conventional specification in which β multiplies future utility. A sufficient condition for maxβ∈(0,1] Q(β)(1/β − 1) to exist is that Q(·) is continuous and limβ→0 Q(β)/β = 0. 6

Issuer n’s cost of managing a consumer’s account is cn < a, and all issuers acquire funds at an interest rate of zero. In period 0, each issuer chooses salient charges (such as annual fees) fn ∈ R to be paid in period 3, and a gross interest rate Rn ≥ 1. Simultaneously with its pricing decision, each firm decides whether to unshroud the costs associated with credit-card use. If no firm unshrouds, consumers assume that they have β = 1. If a firm unshrouds, consumers learn that their β’s are drawn from Q(·), but they do not learn their individual β’s. Observing firms’ offers, in period 0 a consumer decides whether to get a credit card, valuing its future convenience use at v, and the outside option (such as getting a debit card) at u. A consumer can get multiple cards, but needs only one for convenience use, and gets other card(s) only if she sees a strict further benefit. If a consumer gets at least one credit card, she runs up charges of 1 in period 1.6 If a consumer is indifferent between multiple credit-card deals, she lexicographically applies an exogenously given preference over cards to decide which one to get and charge.7 If she charges card n, she decides in period 2 whether to repay 1 to firm n immediately, or repay Rn in period 3.8 A unit of credit-card spending generates a unit of instantaneous utility, and repaying a unit of outstanding debt costs a unit of instantaneous utility. We now show that the above credit-card pricing game simplifies to our reduced-form model. To facilitate the statement, we say that outcomes in two Nash equilibria are the same if the two equilibria differ at most in prices at which consumers do not buy. Lemma 1. The sets of Nash-equilibrium outcomes when fn is unrestricted (fn ∈ R) and when 6 We assume that the consumer makes no charging decision. This makes little difference to the logic of our results. Since a naive consumer does not anticipate that she will delay repayment, she is happy to charge her card for its convenience benefit, and does not care about the interest rate on her card. Furthermore, an endogenous charging decision makes unshrouding less desirable for a firm, since a consumer who is aware that she might pay interest can avoid this by not using her card. Finally, for the same reason as below, a firm would not want to offer a negative up-front price when shrouding occurs. 7 This assumption has the stark implication that if a competitor gets a consumer to sign up for a second, less preferred card, it gets no interest payments from the consumer. This implication plays a central role in our proof that there is a price floor. But a price floor can arise whenever consumers with multiple cards split their expenditures across cards. All that is necessary is that the cost of serving additional consumers below the price floor is greater than the revenue the firm generates from the charges it attracts from holders of multiple cards. 8 For simplicity, we assume that a consumer does not transfer balances to another credit card in period 2, which— under the assumption that there is a market for transfers—she would prefer to do in our three-period model unless the effort cost of doing so was high. As predicted by O’Donoghue and Rabin (2001) and is consistent with evidence in Shui and Ausubel (2004), however, in a more realistic, long-horizon, setting naive borrowers may procrastinate for a long time before finding or taking advantage of favorable balance-transfer opportunities. 7

fn is restricted to be non-negative (fn ≥ 0) are the same. Furthermore, if fn is restricted to be non-negative, then defining an ≡ Q(1/Rn )(Rn − 1), the game is strategically equivalent to (i.e., for any action profile generates the same payoffs as) our reduced-form model with f = 0. The key observation is that to avoid unprofitable consumers, firms act as if they were facing a floor of zero on the up-front price. Intuitively, although setting fn < 0 rather than fn = 0 induces more consumers to get firm n’s card, it does not prevent them from getting and using an alternative card they prefer. Since firm n’s profits derive from consumers using rather than just getting its card, the consumers it attracts with the negative up-front price are unprofitable. Given that fn ≥ 0 for all n, each consumer gets at most one card. The resulting interest payment on the card, then, acts as an additional price in that it adds to the firm’s profits and lowers the consumer’s utility by the same amount. Furthermore, if shrouding occurs, a consumer expects to repay early, so she expects to pay no additional price, whereas if unshrouding occurs, she fully understands how much she will pay in expectation. This logic also makes clear that beyond interest payments, a late fee or any other unanticipated fee that transfers utility from consumers to firms can also serve as an additional price.9 Notice that in our framework, an issuer prefers to undercut competitors’ non-negative up- front prices if it can require consumers to use only the issuer’s card in exchange, thereby generating interest payments down the line. This pricing strategy is consistent with the observation that many credit cards offer usage-contingent perks such as airline miles and rental-car insurance. But while such a strategy avoids unprofitable naive consumers, it may disproportionately attract unprofitable sophisticated consumers, so that to avoid these consumers firms may again act as if they were facing a price floor. Formally, suppose that a fraction λ of consumers has convenience benefit v 0 < v (e.g., because they like paying in cash more than others), and has β = 1 and hence does not pay interest. If firm n lowers fn below v 0 , it attracts all these sophisticated consumers. If λ(v 0 − cmin ) + (1 − λ)(v 0 + a − cmin ) ≤ 0, therefore, the price cut is unprofitable. 9 Suppose, for example, that each firm n can impose a fee ân ≥ 0 for minor infractions such as paying the bill a few days late. A consumer will pay the fee with probability Q(ân ), but unless unshrouding occurs, at the time of selecting a card she believes this probability will be zero. We suppose that Q(â) is continuous and maxâ Q(â)â exists. Then, an = Q(ân )ân acts like an additional price, with the maximum additional price being ā = maxâ Q(â)â. 8

2.2.2 Mutual Funds As a second important application, we discuss mutual funds. The Investment Company Act of 1940 in the US regulates the institutional structure, including the trading prices and accounting, of mutual funds. A fund must calculate its net asset value (NAV) based on market valuations, and including investment advisory fees and other expenses to date, at least once daily. The price of a share is simply the NAV divided by the number of shares, and investors can buy or redeem shares only at the per-share NAV. In addition, a fund can impose purchase fees (front loads).10 Management fees are subject to a fiduciary duty, which we interpret as putting a legal bound on the management fee.11 While disclosure regulations are in place to make sure investors understand the costs of fund ownership, different types of empirical evidence suggest that investors do not fully understand the management fees, and that they appreciate front loads better.12 Formally, we assume that funds and investors interact over three periods, t = 0, 1, 2. In period 0, funds choose up-front prices (front loads) fn and management fees ân satisfying 0 ≤ ân ≤ Â, where the upper bound comes from fiduciary duty. Simultaneously with this choice, firms make unshrouding decisions. Fund n’s cost is cn per consumer initially investing in the fund. Consumers make investment decisions in period 0, starting off with wealth v. Consumers’ 10 See rules 2a-4 and 22c-1 adopted under the Investment Company Act, or the summaries by the Securities and Exchange Commission available at http://www.sec.gov/answers/nav.htm and http://www.sec.gov/divisions/ investment/icvaluation.htm (accessed October 1, 2014). A fund can also impose redemption fees (rear loads), but since these are not important for our arguments, we do not incorporate them in our model. 11 See Section 36 of the Investment Company Act. The Act does not impose a specific maximum fee. Nevertheless, fiduciary duty means that a manager must act in a way that benefits investors, and a management fee that leads an investor into (say) an almost certain loss clearly violates this restriction. Indeed, some lawsuits have successfully challenged excessive management fees on the basis of violating fiduciary duty: http://www.pionline.com/article/ 20111212/PRINT/312129971/wal-mart-merrill-lynch-settle-401k-fee-suit (accessed October 1, 2014). 12 Using a natural experiment in India, Anagol and Kim (2012) show that mutual-fund investors are less sensitive to amortized “initial issue expenses” than to otherwise identical “entry loads” paid up front. In a laboratory experiment on portfolio choice between S&P 500 index funds with different fees, Choi, Laibson and Madrian (2010) find that subjects, including Wharton MBA students, Harvard undergraduates, and Harvard staff members, overwhelmingly fail to minimize fees. Using administrative data from the privatized Mexican Social Security system, Duarte and Hastings (2012) show that investors were not sensitive to fees in choosing between funds, leading to high management fees despite apparently strong competition. Barber, Odean and Zheng (2005) find a strong negative relationship between mutual-fund flows and front loads, but no relationship between flows and operating expenses, and Wilcox (2003) also presents evidence that the overwhelming majority of consumers overemphasize loads relative to expense ratios. Based on a survey of 2,000 mutual-fund investors, Alexander, Jones and Nigro (1998) report that only 18.9% could provide any estimate of their largest mutual fund’s expenses, although 43% claim that they knew this information at the time of purchase. 9

outside option generates utility u in period 2. We normalize the gross return a fund can achieve to 1, so that fund n’s net return is 1 − ân , but consumers perceive the gross return to be R̃ ≥ 1. If a consumer selects fund n, then with probability Q(ân ), she becomes dissatisfied with managed funds in period 1 and switches to an alternative investment opportunity with a return of 1.13 We suppose that Q(·) is continuous and increasing, and that if investing in the mutual fund is optimal (i.e., v(1 − ân )2 ≥ u), then Q(ân ) = 0. When choosing a mutual fund, consumers are aware of the up-front prices, but unless at least one firm unshrouds, they ignore management fees. Unshrouding reveals the expected lifetime management fees consumers will pay, but—for simplicity—we suppose that conditional on selecting a mutual fund in period 0, consumers do not change their period-1 switching behavior in response.14 We now show that this game is strategically equivalent to one in which firms choose appropri- ately defined additional prices: Lemma 2. Defining ṽ = R̃2 v, an ≡ v [ân + (1 − Q(ân ))(1 − ân )ân ] and a ≡ maxâ∈[0,Â] v[â + (1 − Q(â))(1 − â)â], the mutual-fund model is strategically equivalent to our reduced-form model. The reason is essentially the same as in our credit-card model: the management fees generate profits to firms and lower consumers’ utility by the same amount, and consumers are aware of it if and only if unshrouding occurs. Finally, we discuss the floor on the up-front price. The language of the rules interpreting the Investment Company Act, and in particular the phrase and definition of purchase fees, strongly suggests that these fees are intended to be non-negative, so that fn ≥ 0. Furthermore, the constraint fn ≥ 0 follows directly from the provision of the Investment Company Act aimed at protecting existing consumers from using their shares to attract new consumers. Section 22(a) requires funds to “eliminat[e] or reduc[e] so far as reasonably practicable any dilution of the value of other outstanding 13 This assumption is one simple way to capture the notion that higher management fees generate lower performance, and lower performance—independently of whether a consumer realizes the importance of fees or understands gross returns—can trigger dissatisfaction, redemption, and finding better alternative investment opportunities. We can think of the alternative investment consumers switch to as a low-cost index fund. In Section 4, we endogenize consumers’ outside option, which they may initially choose in period 0, as a low-cost index fund as well, although this outside option may in principle also be different. 14 If v(1 − ân )2 < u and unshrouding occurs, consumers select the superior outside option, so our assumption about period-1 switching behavior is inconsequential. Otherwise, consumers find the mutual fund optimal independently of whether unshrouding occurs, so they refrain from switching. 10

securities” in favor of new consumers. One simple interpretation of this “no-dilution” rule is that new investors cannot be treated better than existing investors. Since both pay the management fees, an existing investor would strictly prefer to be treated as a new investor whenever fn < 0, so fn < 0 violates no dilution. What gives us extra confidence that this interpretation is correct is that the regulations on pricing were adopted as a way to interpret the no-dilution section of the Act. Indeed, consistent with a hard price floor of zero, we are unaware of any mutual fund that offers cash back, discounts, or other financial incentives to attract new investors. 2.2.3 Other Potential Applications Mortgages. For mortgages with changing repayment terms, such as pay-option mortgages, interest- only mortgages, and mortgages with teaser rates, many consumers underestimate the payments they will have to make once an initial teaser period ends.15 In this environment, we think of the consumer’s repayment costs assuming terms do not change as the up-front price and of the increase in payments as the additional price. The additional price is limited by consumers’ ability to walk away from the mortgage when payments increase. As we show in an earlier version of our paper (Heidhues, Kőszegi and Murooka 2014), a floor on the up-front price can result because consumers would find an overly low up-front price suspicious, and—concluding that “there must be a catch”— would not buy. But because we need to know consumers’ beliefs to identify this price floor, it is difficult to tell from primitives where the floor is. Bank accounts and hotels. In these applications, consumers buy a base product—account main- tenance in the case of bank accounts and a room in the case of hotels—and can then buy an add-on—such as costly overdrafting in the case of bank accounts or gambling entertainment in the 15 For instance, the Option Adjustable-Rate Mortgage (ARM) allows borrowers to pay less than the interest for a period, leading to an increase in the amount owed and sharp increases in monthly payments when the mortgage resets to an amortizing payment schedule. See “Interest-Only Mortgage Payments and Payment-Option ARMs—Are They for You?,” information booklet prepared for consumers by the Board of Governors of the Federal Reserve System, available at http://www.federalreserve.gov/pubs/mortgage_interestonly/mortgage_interestonly.pdf. Based on existing evidence (Cruickshank 2000, Woodward and Hall 2012, Gerardi, Goette and Meier 2009, Gurun, Matvos and Seru 2013, Bucks and Pence 2008) it is plausible to assume that consumers have a limited understanding of mortgages in general, and of these mortgages in particular. And some features of these mortgages, such as the Option ARM’s one- or three-month introductory interest rate, seem to serve only the purpose of deceiving borrowers about the product’s cost. 11

case of Las Vegas hotels—whose cost they do not appreciate at the moment of purchase.16 We can think of the base product’s price as the up-front price and of the utility loss from unexpectedly high add-on payments as the additional price. The additional price is limited by consumers’ ex-post demand response to add-on prices. Banks likely face a floor on the up-front price close to zero for the same reason as do credit cards, but it is unclear whether hotels face a binding price floor. Printers. In the printer market, consumers do not anticipate high cartridge prices.17 To model this situation, we think of the printer price as the up-front price, and suppose that consumers value a printer explicitly or implicitly assuming that cartridges are free. Then, the additional price is consumers’ loss (and firms’ gain) from positive cartridge prices. A price floor may arise from suspicion or other sources, but we do not know whether it is binding. 3 Profitable Deception This section analyzes our basic model, and discusses key economic implications. 3.1 Benchmarks: Non-Binding Price Floor or Sophisticated Consumers First, we characterize equilibrium outcomes when the floor on the up-front price is not binding. In our setting, this means that (even adding the maximum additional price) a firm cannot make profits if it chooses an up-front price equal to the floor. Proposition 1 (Equilibrium with Non-Binding Price Floor). Suppose f ≤ cmin − a. For any ψ ∈ [0, 1], there exist equilibria in which unshrouding occurs with probability ψ. If shrouding occurs in an equilibrium, firms earn zero profits, and consumers buy the product from a most-efficient firm, pay f = cmin − a, a = a, and get utility v − cmin . If unshrouding occurs in an equilibrium, the Bertrand outcome obtains. 16 For more detailed arguments that these products can be interpreted as deceptive, see Armstrong and Vickers (2012) and Gabaix and Laibson (2006), respectively. 17 For instance, Hall (1997) and the UK’s Office of Fair Trading report that a large majority of consumers buying a printer do not know the price of a cartridge. Gabaix and Laibson (2006) invoke cartridge prices as one of their prime examples of hidden fees. 12

Proposition 1 implies that if the price floor is not binding, there is always a deceptive equilib- rium. Since in a deceptive equilibrium consumers do not take into account additional prices when choosing a product, firms set the highest possible additional price, making existing consumers valu- able. Similarly to the logic of Lal and Matutes’s (1994) loss-leader model as well as that of many switching-cost (Farrell and Klemperer 2007) and behavioral-economics theories, firms compete ag- gressively for these valuable consumers ex ante, and bid down the up-front price until they eliminate net profits. In addition, since with these prices a firm cannot profitably undercut competitors, no firm has an incentive to unshroud. By Proposition 1, any equilibrium outcome is identical either to that in the above deceptive equilibrium, or to that in an unshrouded-prices equilibrium, in which unshrouding occurs with probability one. If unshrouding occurs, consumers make purchase decisions based on the total price, so firms effectively play a Bertrand price-competition game, leading to a Bertrand outcome. As a second benchmark, we note what happens when all consumers are sophisticated in that they observe the total prices fn + an and make purchase decisions based on these prices, and also understand that the total value of a product is v. Then, we have standard Bertrand competition in the total price, so standard arguments imply that the Bertrand outcome obtains. For this to be the case, our assumption that f ≤ cmin is crucial. 3.2 Naive Consumers with a Binding Price Floor Taken together, the benchmarks above imply that for profitable deception to occur, both naive consumers must be present and the price floor must be binding. We turn to analyzing our model when this is the case, assuming for the rest of the section that all consumers are naive and f > cn −a for all n. This condition means that setting an up-front price equal to the floor does not preclude a firm from making profits on its consumers. We first identify sufficient conditions for a deceptive equilibrium to exist. As above, in such an equilibrium all firms set the maximum additional price a. Then, since firms are making positive profits and hence have an incentive to attract consumers, they bid down the up-front price to f . With consumers being indifferent between firms, firm n gets market share sn and therefore earns a 13

profit of sn (f + a − cn ). For this to be an equilibrium, no firm should want to unshroud additional prices. If unshrouding occurs, consumers are willing to pay exactly v − u for the product, so firm n can make profits of at most v − u − cn by unshrouding. Hence, unshrouding is unprofitable for firm n if the following “Shrouding Condition” holds: sn (f + a − cn ) ≥ v − u − cn . (SC) A deceptive equilibrium exists if (SC) holds for all n. Furthermore, since sn < 1, (SC) implies that f + a > v − u, so that in a deceptive equilibrium consumers are worse off than with their outside option. Proposition 2 summarizes this result, and characterizes equilibria fully:18 Proposition 2 (Equilibrium with Binding Price Floor). Suppose f > cn − a for all n. I. If (SC) holds with a strict inequality for all n, there is a deceptive equilibrium, in which all firms offer the contract (fn , an ) = (f , a) with probability one, consumers receive utility less than their outside option, and firms earn positive profits. In any other equilibrium, unshrouding occurs with probability one and the Bertrand outcome obtains. II. If (SC) is violated for some n, unshrouding occurs with probability one and the Bertrand outcome obtains. The intuition for why firms might earn positive profits despite facing Bertrand-type price com- petition is in two parts. First, as in previous models and as in our model with a non-binding price floor, firms make positive profits from the additional price, and to obtain these ex-post profits each firm wants to compete for consumers by offering better up-front terms. But once firms hit the price floor, they exhaust this form of competition without dissipating all ex-post profits. Second, since a firm cannot compete further on the up-front price, there is pressure for it to compete on the additional price—but this requires unshrouding and is therefore an imperfect substitute for competition in the up-front price. If a firm unshrouds and cuts its additional price by a little bit, a consumer learns not only that the firm’s product is the cheapest, but also that all products are more expensive than she thought. If the consumer receives negative surplus from 18 We ignore the knife-edge case in which no firm violates (SC) but (SC) holds with equality for some firm n0 . The equilibrium outcomes characterized in Case I of Proposition 2 remain equilibrium outcomes in this case, but there can be additional equilibrium outcomes. 14

buying at the current total price (i.e., if f + a > v − u), this surprise leads her not to buy, so the firm can attract her by unshrouding only if it cuts the additional price by a discrete margin. Since this may be unprofitable, the firm may prefer not to unshroud. As the flip side of the above logic, if purchasing at the total market price is optimal (i.e., if f + a ≤ v − u), then despite the negative surprise a consumer is willing to buy from a firm that unshrouds and undercuts competitors’ additional prices by a little bit, so in this case a deceptive equilibrium does not exist. Our model therefore says that deception—and positive profits for firms—must be associated with suboptimal consumer purchase decisions.19 Part II of Proposition 2 establishes that (SC) is not only sufficient, but also necessary for profitable deception to occur. To understand the rough logic, assume toward a contradiction that all firms shroud with positive probability. Then, a firm can ensure positive profits by shrouding and choosing prices (f , a), so that in equilibrium firms earn positive expected profits. Since a firm that sets the highest total price when unshrouding has zero market share if some other firm unshrouds, to earn positive profits it must be that with positive probability all rivals set higher total prices when shrouding. Whenever a firm sets one of these high total prices, the only event in which it can make profits is when shrouding occurs, so we can think of its incentives by conditioning on this event. Doing so, arguments akin to those above imply that in this high range firms set prices (f , a), and hence firm n earns sn (f + a − cn ). But firm n can earn v − u − cn by unshrouding, so a firm that violates (SC) prefers to unshroud. Note that if a consumer learns about the additional price before paying any of it and can at that point switch without incurring any cost, then a ≤ v − u must hold. This implies that if in addition f = 0, (SC) is violated, so a deceptive equilibrium cannot occur. In our main applications, however, these prerequisites for (SC) to fail are violated, and in particular there is no consideration 19 Part I of Proposition 2 also states that whenever a deceptive equilibrium exists, there is also an equilibrium in which unshrouding occurs with probability one and—because prices are then transparent and the standard Bertrand logic holds—the Bertrand outcome obtains. As in the case of a non-binding price floor (Proposition 1), since un- shrouding is costless, if a firm unshrouds it is (weakly) optimal for other firms to unshroud as well. Unlike in the case of a non-binding price floor, however, there is no equilibrium in which unshrouding occurs with an interior probability. Our proof of this claim elaborates on the following contradiction argument. Suppose that shrouding occurs with an interior probability, and consider a firm that unshrouds with positive probability and sets the highest price of any firm conditional on unshrouding. When setting this highest price, the firm earns profits only if all other firms shroud, and even then it earns at most v − u − cn . If it instead shrouds and sets (f , a), then it earns at least sn (f + an − cn ) if all other firms also shroud. By (SC), therefore, the firm prefers to shroud with probability 1, a contradiction. 15

that limits the potential size of a relative to v − u. For credit cards, the consumer accumulates debt before she realizes her true β, so she cannot walk away without consequences. For mutual funds, the consumer pays management fees before realizing it. More generally, in many or most applications, going back to the pre-contracting situation is quite costly. In the next subsection, we analyze in more detail circumstances under which profitable deception does versus does not occur. 3.3 Key Economic Implications We first distinguish two cases according to whether the product is socially valuable or wasteful. Non-vanishingly socially valuable product (there is an > 0 such that v − u > cn + for all n). In this case, the right-hand side of (SC) is positive and bounded away from zero. Hence, a firm with a sufficiently low sn violates (SC)—since it earns low profits from deception, it prefers to attract consumers through unshrouding—and thereby guarantees that only a zero-profit unshrouded-prices equilibrium exists. This implies that a deceptive equilibrium can exist if the number of firms is sufficiently small, but not if the number of firms is large—as some firm will then violate (SC). Furthermore, an increase in the number of firms can induce a regime shift from a high-price, deceptive equilibrium to a low-price, transparent equilibrium.20 Socially wasteful product (v − u < cn for all n). In this case, the right-hand side of (SC) is negative while the left-hand side is positive. Hence, a deceptive equilibrium exists regardless of the industry’s concentration and other parameter values: Corollary 1 (Wasteful Products). Suppose f > cmin − a and v − u < cn for all n. Then, a profitable deceptive equilibrium exists. This perverse result has a simple logic: since a socially wasteful product cannot be profitably sold once consumers understand its total price, a firm can never profit from coming clean. Assuming that the outside option is an appropriately chosen alternative product, our main applications seem to fit this case of our model. A low-cost debit card tied to the consumer’s bank 20 The reason for stating our result for non-vanishingly socially valuable products is that if the social value of a firm’s product (v − u − cn ) could be arbitrarily small, then even a firm with low profits from deception might not be willing to unshroud. A precise condition for when unshrouding must occur is N > (f + a)/. Then, sn < /(f + a) for some n, and for this n we have sn (f + a − cn ) < < v − u − cn , in violation of (SC). 16

account serves the same convenience benefit as a credit card, so it is a superior outside option for many consumers. Consistent with this perspective, Laibson et al. (2007) estimate that owning a credit card makes the average household whose head has a high-school degree but not a college degree worse off by the equivalent of paying $2,000 at age 20. Similarly, although not everyone agrees with this view, many researchers believe that because few mutual-fund managers can persistently and significantly outperform the market, most managed funds are inferior to low-cost index funds.21 And while a sharply increasing mortgage-payment schedule makes sense for the few consumers who can look forward to drastic increases in income (as argued by Cocco 2013), the vast majority in this market would have been better served by traditional mortgages. Our model says that such products are sold and remain profitable in a seemingly competitive market not despite, but exactly because they are socially inferior to alternatives. Indeed, the pricing structure of credit cards and mutual funds is consistent with our deceptive equilibrium. DellaVigna and Malmendier (2004) document that most of the popular credit cards in the US offer a $0 annual fee and a high regular interest rate.22 And many mutual funds have zero salient front loads, but substantial management fees.23 The different logic of socially valuable and socially wasteful industries in our model yields two potentially important further points. First, our theory implies that if an industry experiences a lot of entry and does not come clean in its practices, it is likely to be a socially wasteful industry. Second, our theory suggests a general competition-impairing feature in valuable industries that is not present in wasteful industries: to reduce the motive to deviate from their preferred positive- 21 See, for example, Gruber (1996), Carhart (1997), Kosowski, Timmermann, Wermers and White (2006), and French (2008). Berk and Green (2004) and Berk and van Binsbergen (2012), however, argue that the evidence is consistent with the hypothesis that most managers can outperform the market, and that managed funds are not inferior to index funds. 22 There are two main categories of exceptions to the $0 annual fee. First, charge cards—which require repayment within 30 days—often have an annual fee. This is consistent with the prediction of our multi-product model below, where sophisticated consumers who anticipate paying costly interest prefer to use a charge card with a higher up-front price to a credit card with a lower up-front price. Second, some cards offered to consumers with bad credit have an annual fee. This presumably compensates issuers for credit risk, a consideration that is outside our model. 23 In the US market, our model best fits level-load and no-load mutual fund shares ($6,748 billion in total net assets in 2012 according to the ICI fact book 2013, Figure 5.11), which generally have zero front loads and hence face a binding floor on the up-front price. Front-end load shares ($1881 billion in total net assets in 2012) have positive front loads, and are therefore a less obvious fit for our model with a binding price floor. But the typical such fund uses the front load largely to pay intermediaries’ commissions, so the up-front price collected by the fund is often at zero and hence again at the binding floor. 17

profit deceptive equilibrium in a valuable industry, each firm wants to make sure competitors earn sufficient profits from shrouding. This feature is likely to have many implications beyond the current paper (as, for instance, for innovation incentives in Heidhues et al. 2012), and implies that wasteful industries may sometimes be more fiercely competitive than valuable ones. Misprediction of value. We now turn to some implications of Proposition 2 for consumer mis- prediction of value. An economically relevant possibility is one where consumers underestimate the value by a significant amount. Corollary 2 (Unanticipated Valuable Add-On). If v − ṽ ≥ a, then in any equilibrium unshrouding occurs with probability one. The condition v − ṽ ≥ a holds if consumers are completely unaware of a valuable add-on that they can purchase after getting the product, and the outside option does not have such add-ons. In this case, a deceptive equilibrium does not exist. Intuitively, if consumers are unaware of something they would be happy to pay for, then unshrouding cannot make them less willing to buy the product, so unshrouding and undercutting competitors is a profitable deviation from a candidate deceptive equilibrium. For instance, if consumers are unaware of some valuable features of smartphones, producers have all the incentive to explain them. Corollary 2 has a potentially important implication regarding the interpretation of unantici- pated borrowing in the credit-card market. Suppose that—unlike in our model with naive time- inconsistent borrowers—unanticipated borrowing generates unanticipated value for a consumer, for instance by facilitating an unexpected important expense.24 Then, Corollary 2 implies that un- shrouding should take place. From the perspective of our model, therefore, the observation that unshrouding has not taken place implies that expensive credit-card borrowing is not only unantic- ipated, but—consistent with a time-inconsistent setup—also initially unwanted by consumers. Beyond time inconsistency, our applications identify two further reasons for an unanticipated payment not to go along with unanticipated product value. First, consumers may simply fail to 24 Formally, we can modify our model in Section 2.2 by assuming that the consumer is time consistent, values the convenience benefit of the card at ṽ, and can in period 2 spend 1 on something she values at 1 + v − ṽ. She does not anticipate this need ex ante, and if she decides to spend, she cannot repay her credit-card balance. This implies that she is willing to pay at most a = v − ṽ to delay repayment. 18

anticipate a charge—such as a late fee for credit cards or management fee for mutual funds—that is imposed without creating any additional value. Second, consumers may understand that there is a valuable add-on—e.g., that a printer accommodates cartridges for printing—but explicitly or implicitly underestimate the add-on price or not think about add-on expenses when buying the base product. Now suppose that consumers overestimate the product’s value (ṽ > v). Holding other parame- ters constant, this does not affect (SC), and hence does not affect whether a deceptive equilibrium exists in our model. Going slightly beyond our assumptions, however, consumers’ overestimation of value can facilitate the creation of a socially wasteful industry by making consumers more willing to buy the product. In particular, if v − f < u < ṽ − f , then a profitable deceptive equilibrium exists when consumers perceive the value to be ṽ, but not when they perceive the value to be v. For instance, the popularity of managed mutual funds is probably due not only to consumers’ underestimation of their expenses, but also to consumers’ overestimation of their gross returns. 4 Sophisticated Consumers and Multi-Product Markets In this section, we incorporate sophisticated consumers—who observe but cannot avoid additional prices—into our model, considering both single- and multi-product markets. We assume that the proportion of sophisticated consumers is κ ∈ (0, 1), that the price floor is binding (f > cn − a for all n), and that consumers perceive the product’s value correctly (ṽ = v). 4.1 Sophisticated Consumers in Our Basic Model Notice that in any deceptive equilibrium, sophisticated consumers do not buy the product: if they did, they would buy from a firm with the lowest total price, and such a firm would prefer to either undercut equal-priced competitors and attract sophisticated consumers, or (if there are no equal- priced competitors) to unshroud and attract all consumers. Hence, the total price of any firm exceeds v − u. This implies that if firm n unshrouds, it maximizes profits by setting a total price equal to v − u, attracting all naive and sophisticated consumers. Combining these considerations, 19

unshrouding is unprofitable if (1 − κ)sn (f + a − cn ) ≥ v − u − cn , (1) and a deceptive equilibrium exists if and only if Condition (1) holds for all n. If the product is socially valuable, the condition for a deceptive equilibrium to exist is stricter if sophisticated consumers are present than if they are not. Intuitively, while sophisticated consumers do not buy the product when the additional price is high, they can be attracted by a price cut, creating pressure to cut the additional price—and by implication also to unshroud. This result is consistent with the insights of Gabaix and Laibson (2006) and Armstrong and Vickers (2012) that if the proportion of sophisticated consumers is sufficiently high, transparency and efficiency obtains. But if the product is socially wasteful, then as before a profitable deceptive equilibrium always exists. The reason is simple: sophisticated consumers never buy a socially wasteful product in equilibrium, so their presence is irrelevant—firms just exploit naive consumers. 4.2 Sophisticated Consumers with an Alternative Transparent Product In many markets for deceptive products, alternatives that are more transparent than and arguably superior to the deceptive products exist. Motivated by this observation, we modify our model above by assuming that each firm has a transparent product in addition to the potentially deceptive product we have described. Because we think of the alternative product as an endogenous outside option, we set consumers’ utility from not buying, u, to zero. Consumers’ value for the transparent product is w > 0, and firm n’s cost of producing it is cw n ≥ 0, with at least two firms having the lowest cost cw w w min = minn {cn }. Crucially, we posit that product w is socially valuable (w −cmin > 0), and is not inferior to product v: w − cw min ≥ v − cmin . Consumers are interested in buying at most one product. Firms simultaneously set the up-front and additional prices for product v, the single transparent price for product w, and decide whether to unshroud the additional prices of product v. If consumers weakly prefer buying and are indifferent between a number of firms in the market for product v or w, firms split the respective market in proportion to sn or sw n , respectively; and if consumers are indifferent between products v and w, a given positive fraction of them chooses 20

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